In environmental and health studies, it is important to detect the beneficial or adverse effects of a new chemical or process as quickly as possible. This need has motivated the investigation of selected statistical procedures for early decision to determine their error probabilities and expected sample sizes. Primary emphasis will be given to curtailed procedures for parametric and nonparametric tests and their possible extensions to sequential procedures. Central limit theorems for large deviations will be used to obtain asymptotic which will compared to small sample results generated by Monte Carlo simulation studies and to sequential procedures. Complete analysis of curtailed one-sample nonparametric tests and curtailed procedures for the one-parameter exponential family under the null hypothesis and alternatives are underway. Extensions to two-sample nonparametric tests, multiparameter distributions and other parametric distributions are planned using similar techniques. Random walk approximations and Monte Carlo studies will be used to develop sequential nonparametric tests. A literature review of testing and estimation from a single data set will also begin.